cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A180783 Number of distinct solutions of Sum_{i=1..1} (x(2i-1)*x(2i)) == 1 (mod n), with x() in {1,2,...,n-1}.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 4, 4, 3, 6, 4, 7, 4, 6, 6, 9, 4, 10, 6, 8, 6, 12, 8, 11, 7, 10, 8, 15, 6, 16, 10, 12, 9, 14, 8, 19, 10, 14, 12, 21, 8, 22, 12, 14, 12, 24, 12, 22, 11, 18, 14, 27, 10, 22, 16, 20, 15, 30, 12, 31, 16, 20, 18, 26, 12, 34, 18, 24, 14, 36, 16, 37, 19, 22, 20, 32, 14, 40, 20, 28
Offset: 1

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Author

R. H. Hardin, formula from Max Alekseyev in the Sequence Fans Mailing List, Sep 20 2010

Keywords

Comments

Except for the first term, this appears to be the number of pairs of integers i,j with 1 <= i <= n, 1 <= j <= i, such that i+j == i*j (mod n), for n=1,2,3,... - John W. Layman, Oct 19 2011
Layman's observation holds since i+j == i*j (mod n) is equivalent to (i-1)*(j-1) == 1 (mod n). - Max Alekseyev, Oct 22 2011
For i > 1, equal to the number of elements x relatively prime to n such that x mod n >= x^(-1) mod n. - Jeffrey Shallit, Jun 14 2018
Differs from A007897 for n = 1, 35, 45 etc. - Georg Fischer, Sep 20 2020

Examples

			Solutions for product of a single 1..10 pair = 1 (mod 11) are (1*1) (2*6) (3*4) (5*9) (7*8) (10*10).

Links

Crossrefs

Column 1 of A180793.
Cf. A000010, A007897, A060594.

Formula

a(n) = (A000010(n) + A060594(n)) / 2.

A180784 Number of distinct solutions of sum{i=1..2}(x(2i-1)*x(2i)) = 1 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 0, 1, 4, 9, 19, 31, 42, 75, 91, 136, 160, 232, 254, 364, 388, 542, 525, 767, 754, 1015, 993, 1389, 1256, 1795, 1641, 2169, 2080, 2838, 2344, 3484, 3144, 3971, 3676, 4980, 4152, 5989, 5135, 6564, 6008, 8195, 6392, 9476, 8064, 9912, 9114, 12426, 9808, 14032
Offset: 1

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Author

R. H. Hardin Sep 20 2010

Keywords

Comments

Column 2 of A180793

Examples

			Solutions for sum of products of 2 1..5 pairs = 1 (mod 6) are
(1*1 + 2*3) (1*1 + 3*4) (1*2 + 1*5) (1*3 + 1*4) (1*3 + 2*2) (1*3 + 2*5)
(1*3 + 4*4) (1*4 + 3*3) (1*4 + 3*5) (1*5 + 2*4) (1*5 + 4*5) (2*2 + 3*3)
(2*2 + 3*5) (2*3 + 5*5) (2*5 + 3*3) (2*5 + 3*5) (3*3 + 4*4) (3*4 + 5*5)
(3*5 + 4*4)

Links

A180785 Number of distinct solutions of sum{i=1..3}(x(2i-1)*x(2i)) = 1 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 1, 3, 14, 46, 106, 254, 494, 939, 1528, 2668, 3958, 6334, 8641, 13239, 17240, 25227, 31128, 44660, 53786, 74111, 86712, 118779, 134632, 181755, 202431, 266175, 296323, 387533, 412008, 544121, 582432, 736568, 786710, 1006750, 1037604, 1336781
Offset: 1

Views

Author

R. H. Hardin Sep 20 2010

Keywords

Comments

Column 3 of A180793

Examples

			Solutions for sum of products of 3 1..3 pairs = 1 (mod 4) are
(1*1 + 1*1 + 1*3) (1*1 + 1*2 + 1*2) (1*1 + 1*2 + 2*3) (1*1 + 1*3 + 3*3)
(1*1 + 2*2 + 2*2) (1*1 + 2*3 + 2*3) (1*2 + 1*2 + 3*3) (1*2 + 1*3 + 2*2)
(1*2 + 2*3 + 3*3) (1*3 + 1*3 + 1*3) (1*3 + 2*2 + 2*3) (1*3 + 3*3 + 3*3)
(2*2 + 2*2 + 3*3) (2*3 + 2*3 + 3*3)

Links

A180786 Number of distinct solutions of sum{i=1..4}(x(2i-1)*x(2i)) = 1 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 0, 7, 30, 142, 502, 1519, 3828, 9145, 18966, 38562, 70202, 127954, 211261, 357465, 549988, 875942, 1273587, 1941522, 2705012, 3966472, 5325916, 7599591, 9892052, 13772034, 17476435, 23770735, 29625591, 39617904, 48129046, 63654183, 76396024
Offset: 1

Views

Author

R. H. Hardin Sep 20 2010

Keywords

Comments

Column 4 of A180793

Examples

			Solutions for sum of products of 4 1..2 pairs = 1 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*1) (1*1 + 1*1 + 1*1 + 2*2) (1*1 + 1*1 + 2*2 + 2*2)
(1*1 + 1*2 + 1*2 + 1*2) (1*1 + 2*2 + 2*2 + 2*2) (1*2 + 1*2 + 1*2 + 2*2)
(2*2 + 2*2 + 2*2 + 2*2)

Links

A180787 Number of distinct solutions of sum{i=1..5}(x(2i-1)*x(2i)) = 1 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 1, 5, 64, 400, 1914, 7589, 24902, 73134, 188315, 455127, 996083, 2098908, 4080930, 7790731, 13869018, 24527424, 40768032, 67955336, 106952861, 169909977, 255608028, 390623704, 566247614, 837326375, 1175991196, 1690928973
Offset: 1

Views

Author

R. H. Hardin Sep 20 2010

Keywords

Comments

Column 5 of A180793

Examples

			Solutions for sum of products of 5 1..2 pairs = 1 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*2 + 1*2) (1*1 + 1*1 + 1*2 + 1*2 + 2*2)
(1*1 + 1*2 + 1*2 + 2*2 + 2*2) (1*2 + 1*2 + 1*2 + 1*2 + 1*2)
(1*2 + 1*2 + 2*2 + 2*2 + 2*2)

Links

A180788 Number of distinct solutions of sum{i=1..6}(x(2i-1)*x(2i)) = 1 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 0, 9, 112, 1004, 6404, 32890, 137528, 499641, 1579113, 4551268, 11861187, 29034355, 65777365, 142805210, 291148080, 576393509, 1082580072, 1993354411, 3505999065, 6088877416, 10148269838, 16796812567, 26776385524, 42563703291
Offset: 1

Views

Author

R. H. Hardin Sep 20 2010

Keywords

Comments

Column 6 of A180793

Examples

			Solutions for sum of products of 6 1..2 pairs = 1 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2) (1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*2 + 2*2 + 2*2) (1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2)
(1*1 + 1*1 + 1*2 + 2*2 + 2*2 + 2*2) (1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2)
(1*1 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2) (1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2)
(1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)

Links

A180789 Number of distinct solutions of sum{i=1..7}(x(2i-1)*x(2i)) = 1 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 1, 15, 198, 2282, 19300, 126861, 670058, 2997685, 11539243, 39660969, 122371876, 348412793, 914595808, 2264326584, 5259342780, 11692554312, 24683815072, 50403390786, 98560661538, 187881799209, 345060981679, 621482071341
Offset: 1

Views

Author

R. H. Hardin Sep 20 2010

Keywords

Comments

Column 7 of A180793

Examples

			Solutions for sum of products of 7 1..2 pairs = 1 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 2*2)
(1*1 + 1*1 + 1*1 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2)
(1*1 + 1*1 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2)
(1*1 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2)
(1*1 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2)
(1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2)
(2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)

Links

A180790 Number of distinct solutions of sum{i=1..8}(x(2i-1)*x(2i)) = 1 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 0, 12, 318, 4868, 53133, 444019, 2935676, 16112283, 75118525, 307372602, 1118328968, 3701885454, 11222027568, 31699803817, 83633494240, 209004408969, 494557127475, 1121475446283, 2431614016920, 5096216430113, 10285962253471
Offset: 1

Views

Author

R. H. Hardin Sep 20 2010

Keywords

Comments

Column 8 of A180793

Examples

			Solutions for sum of products of 8 1..2 pairs = 1 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2)
(1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2)
(1*1 + 1*1 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2)
(1*1 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2)
(1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2)
(1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)

Links

A180791 Number of distinct solutions of sum{i=1..9}(x(2i-1)*x(2i)) = 1 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 1, 18, 502, 9721, 135902, 1430707, 11752870, 78770466, 442714845, 2151608163, 9202214058, 35373572486, 123540794708, 398005342048, 1190430659772, 3344070542314, 8854626554163, 22304900540858, 53543330420874
Offset: 1

Views

Author

R. H. Hardin Sep 20 2010

Keywords

Comments

Column 9 of A180793

Examples

			Solutions for sum of products of 9 1..2 pairs = 1 (mod 3) are
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*1 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*1 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2)
(1*1 + 1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*1 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2)
(1*1 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*1 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2)
(1*2 + 1*2 + 1*2 + 1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)
(1*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2 + 2*2)

Links

A180792 Number of distinct solutions of sum{i=1..10}(x(2i-1)*x(2i)) = 1 (mod n), with x() only in 1..n-1.

Original entry on oeis.org

0, 0, 26, 744, 18475, 326446, 4292149, 43506488, 354466019, 2391684440, 13770292256, 69044240450, 307750080388, 1235903232638, 4537248102720, 15362780912768, 48489022864600, 143504551536641, 401488209735955
Offset: 1

Views

Author

R. H. Hardin Sep 20 2010

Keywords

Comments

Column 10 of A180793

Links

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